To determine the maximum length possible for Charlene's baby blanket, we need to consider the given conditions and solve the system of inequalities step-by-step.
The inequalities given are:
1. [tex]\( w \geq 0.5l \)[/tex] (The width must be at least half the length)
2. [tex]\( 2l + 2w \leq 180 \)[/tex] (The perimeter of the blanket must not exceed 180 inches)
Let's walk through the solution:
1. Simplify the Perimeter Inequality:
[tex]\[ 2l + 2w \leq 180 \][/tex]
We can simplify this by dividing every term by 2:
[tex]\[ l + w \leq 90 \][/tex]
2. Substitute the Width Constraint:
From the first inequality, we know:
[tex]\[ w \geq 0.5l \][/tex]
To find the maximum length [tex]\( l \)[/tex], we replace [tex]\( w \)[/tex] with the smallest possible value given by [tex]\( 0.5l \)[/tex]. So:
[tex]\[ l + 0.5l \leq 90 \][/tex]
3. Combine Like Terms:
Combine the terms on the left side:
[tex]\[ 1.5l \leq 90 \][/tex]
4. Solve for [tex]\( l \)[/tex]:
Divide both sides by 1.5 to isolate [tex]\( l \)[/tex]:
[tex]\[ l \leq 60 \][/tex]
This tells us that the maximum length [tex]\( l \)[/tex] can be is 60 inches.
To ensure this length satisfies all conditions, we calculate the corresponding width [tex]\( w \)[/tex]:
[tex]\[ w = 0.5 \times 60 \][/tex]
[tex]\[ w = 30 \][/tex]
Checking the perimeter:
[tex]\[ 2l + 2w = 2 \times 60 + 2 \times 30 = 120 + 60 = 180 \][/tex]
Thus, the perimeter condition is satisfied.
Therefore, the maximum possible length for Charlene's baby blanket is:
[tex]\[ l = 60 \text{ inches} \][/tex]
So, the correct answer is:
[tex]\[
\boxed{60 \text{ inches}}
\][/tex]
